---
layout: post
title: Stress and strains in soil mechanics
date: Tue 16 Mar 2021 05:26:21 PM IST
categories: ["Academic Writings", "Soil Mechanics and Foundations"]
---
-
1 Introduction
We design the substructures to avoid excessive settlement and collapse of the foundation soil. Substructures transfer the load from superstructure to the soil. We assume soil as continua and calculate the stress and strain field by solving the equilibrium equations. This requires constitutive relations for the soils–stress-strain behavior under different conditions.
A mathematical model serves as the constitutive relation for the soil. We design the experiments to understand the soil and propose a mathematical model based on the experimental observations accounting for physics or statistics of the soil behavior. Experimental modeling consists of elemental experiments, mimicking the behavior of representative soil element, and small scale physical experiments, mimicking a boundary value problem. The elemental experiments help in formulating a constitutive model, whereas physical model tests validate their performance with a numerical model.
This text overviews these aspects in the following.
Consider the continua subjected under different boundary conditions. We define the traction at point on a plane with normal as:
Where is the time, is the coordinate of the point, and is the force acting on an infinitesimal area . The changes to for soils with area of representative volume element as .
We implicitly assume the dependence of time and avoid writing it for the sake of clarity.
Figure 1: Cauchy tetrahedron
2.1 Cauchy’s theorem for the existence of stress
Consider an infinitesimal tetrahedron (size with vertex at , fig. 1) and apply balance of linear momentum:
With infinitesimal tetrahedron, as ,
Using the above:
Now, substitute with in the above:
Since the choice of is arbitrary:
Which is the Newton’s law of action and reaction.
We can write the traction on plane normal to as:
The Cauchy stress is defined as:
The Cauchy stress tensor (at point ) maps the normal (on a given plane through point ) to traction (acts at point , on plane normal to ):
2.2 Stress components
We often deal with stress components () than the stress tensor. The first index represents the plane and second index represents the direction in which it acts. For example, represents component of the stress tensor which acts on plane normal to along the direction . If both indices are equal, it represents the normal stress, and different indices represent shear stress.
Figure 2: Stress components
2.3 Balance of linear momentum
Net force acting on a spatial region is equal to its rate of change of linear momentum.
Using the Cauchy’s theorem and Reynolds transport theorem:
Using Gauss’s divergence rule: The region is arbitrary, this leads to
2.4 Balance of angular momentum
Net moment of a spatial region is equal to its rate of change of angular momentum.
Using the Cauchy’s theorem, balance of linear momentum and Gauss’s divergence rule:
We have to apt for the indicial notation:
Since is arbitrary:
Now we have:
Hint: In eq. 1:
Put and only feasible options for are (2,3) and (3,2).
Put and only feasible options for are (1,3) and (3,1).
Put and only feasible options for are (2,1) and (1,2). ## Balance of power
Conventional external power:
Using Cauchy’s theorem and balance of linear momentum:
Considering the above:
We have:
The external power is expended on the internal power and the change in kinetic energy.
2.5 The first law of thermodynamics
The external power expended on and net heat flow into results in increase of net internal energy and kinetic energy.
We can write net internal energy and net heat flow as:
Using the mechanical power balance, we can write the first law of energy as:
2.6 The second law of thermodynamics
The net entropy production in is non-negative.
The rate of change of net internal-entropy is sum of entropy flow and net entropy production in .
The second law of thermodynamics states:
We can write the net internal entropy in terms of the specific entropy :
and entropy flow in terms of entropy flux and entropy supply :
We can rewrite the second law of thermodynamics as:
We have the following hypothesis: Where is the temperature.
Considering the above, we have ``Clausius-Duhem inequality"
Stresses result in deformation; fig. 3 shows an initial configuration and corresponding deformed configuration.
Figure 3: Deformation
fig. 3 shows the initial and deformed configuration of a spatial region . The mapping denotes the motion of these configuration. At a fixed time, this mapping denotes the deformation. We have the following mapping for line element in the initial and deformed configuration.
Mapping of the area element:
Mapping of the volume element:
Definition of strains:
Where is the Green-Lagrangian strain tensor.
Where is the Eulerian-Almansi strain tensor.
The displacement field is:
Now, the strain tensors can be written as:
4 Coordinate transformation
The tensor is invariant to coordinate transform, however, the components changes with the basis. Tensor components transform with the following rule:
We derive this in the following:
Coordinate transformation rule
Stress tensor remains invariant under coordinate transformation
By comparing the components, we have:
4.1 Three dimensional stress tensor
For a plane passing through the point with stress tensor as ,
We evaluate the traction as:
Correspondingly, the normal and shear stress are:
With the above equation for , we have two signs.
We also have
We have the set of linear equations:
Assuming :
With , and assuming :
Figure 4: Three dimensional Mohr circle
fig. 4 shows the admissible stress states.
4.2 Two-dimensional stress tensor
fig. 5 shows the original coordinate system and transformed coordinate system . The rotation matrix is given by:
Figure 5: Coordinate transformation in two-dimension
The stress components in the transformed coordinate system:
If we desire to estimate the normal and shear stress on the plane normal to :
The locus of normal and shear stress is a circle, called Mohr circle.
The Mohr circle is very useful in quickly extracting the tensor values on different planes or in extracting the planes corresponding to different tensor values. The method of poles in particularly interesting in Mohr circles. It only works with the right sign conventions:
Compression positive and counterclockwise shear positive
Tension positive and clockwise shear positive
In the following, we see how pole appears with compression positive and counterclockwise shear positive.
Figure 6: Stress-state
For an element with the stress state given in the fig. 6, we have normal stress and shear sterss on the plane normal to as:
# import the librariesimport numpy as npfrom matplotlib import pyplot as plt# stress state at the points11 =150s12 =50s22 =50# radius and center of the Mohr circleR = np.sqrt(((s11-s22)/2)**2+s12**2)center = (s11+s22)/2phi = np.arctan((s11-s22)/(2*s12)) # twice the inclination of principal plane# number of points on the Mohr circle and lenght of the planenum_points =100length_plane =400# stress states on the Mohr circle for different theta valuessnfn =lambda t: center + R * np.sin(2*t+phi)tnfn =lambda t: -1* R * np.cos(2*t+phi)theta = np.linspace(0, 2*np.pi, num=num_points)sn = snfn(theta)tn = tnfn(theta)'''plot the Mohr circle and the planes on which stresses, at the point in theMohr circle, acts.From [@fig:StressState] direction cosines of the plane are[cos(270+theta), sin(270+theta)] --> [sin(theta), -cos(theta)]The line that represents plane on the Mohr circle has start point$\left(x-\frac{l}{2}\sin{\theta},y+\frac{l}{2}\cos{\theta}\right)$ and endpoint $\left(x+\frac{l}{2}\sin{\theta},y-\frac{l}{2}\cos{\theta}\right)$. Thepoint $\left(x,y\right)$ is the point on the Mohr circle.'''plt.figure()plt.plot(sn, tn)for i in np.linspace(0, 2*np.pi, num =10):# point on the Mohr circle plt.plot(snfn(i), tnfn(i), 'o', MarkerSize=20)# start and end points start = [snfn(i)-length_plane/2*np.sin(i),tnfn(i)+length_plane/2*np.cos(i)] end = [snfn(i)+length_plane/2*np.sin(i),tnfn(i)-length_plane/2*np.cos(i)] point_x = [start[0], end[0]] point_y = [start[1], end[1]] plt.plot(point_x, point_y, '-')plt.axis('equal')plt.xlabel(r'Normal stress $\sigma_{n}$')plt.ylabel(r'Shear stress $\tau_{n}$')
Pole
5.1 Mohr circle of strain
fig. 7 shows the strain state in an element. We consider contraction and counterclockwise shear as positive. In the fig. 7, strain state is the following:
Note that the sign convention here is according the Mohr circle and the above does not imple the strain tensor is assymetric. The engineering shear strain is the change in angle between two perpendicular line elements in the initial configuration.
Figure 7: Strain state
fig. 8 shows the Mohr circle corrsponding to this strain state. The strain components act on the vertical plane; we obtain the pole by drawing the vertical line at which intersects the Mohr circle at the pole. The line joining the pole and zero normal strain points correspond to plane of zero extension and line perpendicular to that is the zero extension line.
Granular media changes its volume upon distortion; this property is known as dilatancy.
Figure 9: Dilatancy
fig. 9 shows an element under distortion along ; there is no contraction or dilation along this direction. The line of zero extension lies along . Angle of dilatancy () is defined as
Slip surfaces in soil mass have the shape of logarithmic spiral. When a contiguous elements reach the failure in soil, a slip surface forms. The soil around the slip surface behaves as a rigid block such that distortion occurs within the slip surface. The tangents to slip surfaces are lines of zero extension. From the fig. 10:
Figure 10: Slip surface
6 Failure criteria
6.1 Mohr-Coulomb failure criteria
Where and are major and minor principal stresses, and , are the peak of the Mohr-Circle.
In a boundary value problem, the principal planes of stresses or strains rotate.
7.1 Finite rotation over a stress discontinuity
7.1.1 Cohesive soil under undrained loading
7.1.1.1 simple case
Figure 11: Stress rotation behind a frictional retaining wall
7.1.1.2 General case
We have a stress discontinuity over a plane which makes angle , in counterclockwise sense, with the vertical. The principal planes are neither horizontal or verical. The Mohr cirlce on both sides of disconituity have the following properties fig. 12.
Figure 12: Stress rotation under general condition
Mean stress
Pole
Plane of discontinuity
Principal plane rotation
–
7.1.2 Cohesion-less soil
Figure 13: Stress rotation in frictional soil
7.2 Infinitesimal rotation over a stress discontinuity
A soil mass fails when a contiguous failure surface forms such that stress state at each point satisfies Mohr-Coulomb failure criteria. An example stress rotation is given in fig. 14.
Figure 14: Stress rotation
For zone A, we can write the following –
For zone B, we can write the following –
In general, the shear and normal stress at the discontinuity can be written as –
At the zone of discontinuity, we also have –
This implies –
7.2.1 Cohesive soils
For the cohesive soil, the Mohr-Coulomb failure criteria is given as:
Where is the undrained cohesive strength.
The radius of Mohr circle doesn’t change at failure –
Note .
7.2.2 Cohesion-less soils
For a cohesion-less soil, the Mohr-Coulomb failure criteria is given as:
Also .
From the above, we have –
8 Earth retaining structures
At rest,
8.1 Frictionless interaction between soil and structure
8.1.1 Rankine theory
8.1.1.1 Flat surface
Figure 15: Active and passive earth pressure for leveled soil mass behind a vertical structure
8.1.1.2 Inclined surface
Figure 16: Active and passive earth pressure for inclined soil mass behind a vertical structure
8.2 Frictional interaction between soil and structure
8.2.1 Coulomb theory
8.2.2 Stress characteristics
9 Readings
A good knowledge of continuum mechanics helps in understanding soil mechanics and soil plasticity.
Gurtin, M.E., Fried, E. and Anand, L., 2010. The mechanics and thermodynamics of continua. Cambridge University Press.